Higgsing Transitions in Gauge Theories

• Higgsing transitions are symmetry-breaking phenomena that reduce the gauge group's rank by driving adjoint nodes to strong coupling and moving the system onto its Coulomb branch.
• They integrate RG flows with duality steps like Seiberg duality, generating extra U(1) factors, moduli, and massless monopoles in the process.
• The dynamics are mapped onto N=2 SQCD and supergravity duals, providing calculable insights into supersymmetry breaking and systematic cascade tracking.

Higgsing transitions are symmetry-breaking phenomena central to the understanding of phase structure in gauge theories, both in four-dimensional field theory and their holographic or string-theoretic realizations. In the context of supersymmetric quiver gauge theories—with nodes that possess adjoint matter—such transitions play a crucial role in duality cascades and renormalization group flows as exhibited in quivers derived from non-isolated singularities, notably in the $\mathbb{Z}_2$ orbifold of the conifold. Adjoint transitions, which occur when an adjoint node is driven to strong coupling, are shown to be dual to Higgsing transitions; they describe a process wherein the rank of a non-Abelian gauge factor is reduced, abelian degrees of freedom and massless moduli are generated, and the system continues to flow toward weaker coupling regimes. Theoretical control over these transitions is achieved by mapping their dynamics to well-established N=2 SQCD behavior, both in field theory and in the supergravity duals of these systems.

1. Adjoint Transitions as Higgsing Duals

Adjoint transitions in quiver gauge theories with supersymmetric matter content are initiated when the gauge coupling of a node with adjoint matter grows strong in the course of an RG flow. At strong coupling, the adjoint sector effectively forces the system onto its Coulomb branch: the vacuum expectation value of the adjoint field spontaneously breaks the non-Abelian gauge group to a subgroup and introduces additional U(1) factors, moduli, and, for special vacua, massless monopoles. This duality between adjoint transitions and Higgsing is explicit, for example, when choosing a diagonal vacuum for the adjoint field as

$$\Phi_{22} = \mathrm{diag}\{0,\ldots,0,\phi_1,\ldots,\phi_{m+k}\}$$

which breaks $U(N)$ to $U(N-k)$, generating extra U(1) factors and moduli parameterizing the Coulomb branch. The formalism treats the strong-coupling regime analogously to N=2 SQCD: the adjoint’s gauge coupling growth is regulated by the shift onto the Coulomb branch. Beta functions for these couplings—e.g.,

$$\beta_{\lambda_2} = \frac{\lambda_2^2 f_2}{8\pi^2}\left[\frac{3P_2}{N} + \mathcal{O}\left(\frac{P^2}{N^2}\right)\right]$$

—demonstrate that couplings are driven to strong values and the theory transitions to a Higgsed phase where the effective rank drops.

2. Mechanism and RG Flow of Higgsing Transitions

The dynamical process of adjoint (Higgsing) transitions is intertwined with RG flows and sequences of dualizations:

• Starting near a conformal fixed point, relevant quartic superpotential couplings (designated $\eta$) become strong, with beta functions

$$\beta_\eta \simeq -\eta \left(\frac{3P_2+3P_4}{2N} + \ldots \right),$$

inducing flows to nonperturbative regions.

• At large $\eta$, Seiberg duality is applied to convert strong quartic interactions into mass terms; integrating out massive mesons reduces the gauge group rank by predictable amounts.
• However, if an adjoint node (say, associated with $\lambda_2$) is then driven to strong coupling, the resulting dynamics cannot be captured by Seiberg duality; the correct description is via N=2 SQCD, in which the Higgsing/Coulomb branch physics induces further rank reduction, introduces U(1) factors, and (when relevant) massless monopoles. Such sequential transitions, alternating between Seiberg duality and Higgsing, structure the entire duality cascade in a way that is calculable.

3. Field Theory Analysis and Supergravity Duals

The paper synthesizes two main technical approaches:

• Field Theory: Detailed beta function computations (via $a$-maximization), the construction of exactly marginal coupling loci (preserving discrete symmetries like $J_-$), and the use of N=2 SQCD structure when adjoint nodes are strongly coupled. The sequence of RG flows interleaves Seiberg dualities and adjoint (Higgsing) transitions, and renders the vacuum structure explicitly in terms of Coulomb branches.
• Supergravity: The gauge theory cascades have direct geometric avatars in warped throat solutions of IIB string theory. Adjoint transitions are mapped to particular radial positions in the throat, where changes occur in the background metric,

$$ds^2 = h(r)^{-1/2} dx^2 + h(r)^{1/2}[dr^2 + (dT_{1,1}/\mathbb{Z}_2)^2],$$

corresponding to the emission (peeling off) of fractional brane sources and corresponding to the appearance of extra light fields in the field theory. Thus, the presence of extra U(1)s, moduli, and monopoles is mirrored by explicit source terms localized in the radial direction.

4. The Role of N=2 SQCD as Regulator

The universal features of these transitions are governed by the N=2 SQCD structure associated with adjoint nodes:

• Nodes with adjoint matter and N=2-like couplings are well-approximated by N=2 gauge theory, whose exact solution reveals an entire Coulomb branch with spontaneous gauge symmetry breaking, additional U(1) factors, moduli, and monopoles.
• Strong coupling at an adjoint node compels the system to choose a vacuum—a point on the Coulomb branch—effectively “Higgsing” the non-Abelian gauge group. The dimension of the Coulomb branch sets the number of emergent U(1) factors.
• The vacuum structure determines the pattern of rank reduction, the number of massless abelian gauge fields, and the presence of additional light states—a mapping that is both explicit and calculable. The free $\mathcal{F}_{l,s}$ sector tracks these degrees of freedom.

5. Implications for Supersymmetry Breaking and Metastable Vacua

Higgsing transitions in cascade scenarios have critical ramifications for the UV completion of metastable supersymmetry-breaking models:

• The appearance of light U(1)s, moduli, and monopoles after adjoint transitions has the potential to contaminate or destabilize the would-be supersymmetry-breaking vacua constructed deep within cascades.
• In warped throat models, integrating the metastable state into a duality cascade unavoidably produces these extra degrees of freedom, which can introduce runaway directions or destabilize the vacuum structure.
• Any viable mechanism for dynamical SUSY breaking in such strongly coupled models must take into account not just the original quiver theory, but the full spectrum of light fields generated through the sequence of Higgsing (adjoint) transitions.

6. Generalization and Prescription for Cascade Tracking

Though analyzed in detail for a $\mathbb{Z}_2$ orbifold of the conifold, the underlying mechanism generalizes:

• Any quiver gauge theory from fractional branes at non-isolated singularities (QNIS) will generically present adjoint nodes with N=2-like couplings.
• The proposed RG-flow prescription is:
  • For a node without adjoints reaching strong coupling, apply Seiberg duality.
  • For a strongly coupled adjoint node, approximate by N=2 SQCD, move the system to an appropriate Coulomb branch vacuum (Higgsing), and integrate out the resulting massive fields to remove the decoupled sector.
  • Recompute the beta functions and proceed with the cascade.
• This approach enables explicit tracking of RG flows in both field theory and holographic descriptions, even into the supergravity regime, providing a systematic mechanism to follow Higgsing transitions in a wide variety of supersymmetric quiver gauge theories.

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The paper of Higgsing transitions in this setting precisely elucidates how duality cascades orchestrate a combination of Seiberg duality steps and adjoint-driven Higgsings, where the rank of the gauge group is reduced in a pattern set by N=2 SQCD Coulomb branch vacua, yielding additional abelian gauge factors, moduli, and monopole degrees of freedom. The interplay between field theoretic RG flows and their supergravity duals leads to a highly controlled understanding of cascade dynamics, with significant consequences for supersymmetry breaking scenarios and model building in the context of both string theory and strongly-coupled field theory (Simic, 2010).